As we know, random variables have few common numerical characteristic – average value, variation and standard deviation. Now we consider standard deviation of the random variable X. Formulas for calculation variation are next:

1. Population variation for random variable, which is defined by sequence \(x_{i}\) :

From these definitions of the variations, we get two definitions of the standard deviation:
σ – population (in some statistical research – known) standard deviation, s – sample (estimated) standard deviation.

Consider a few examples for calculation of the standard deviation of the random variables.

Example 1. Consider the random variable:

Average value for random variable X is next:

\(\overline{x}= 1*0.1+2*0.2+3*0.3+4*0.4=3\)

Using this value, we can calculate variation of the random variable X:

\(\sigma^2 = \frac{1}{n}\sum_{i=1}^n (x_{i}-\overline{x})^2\)

=\({(1-3)}^2*0.1+{(2-3)}^2*0.2+{(3-3)}^2*0.3+{(4-3)}^2*0.4=1\)

From this result, we obtain that standard deviation for random variable X is σ =\(\sqrt{1}\)=1.

3. Let X and Y, two independent random variable, with standard deviations \(σ_{X}\) and \(σ_{Y}\) respectively, then standard deviation of the sum and difference of these random variables are the same and:

\(\sigma_{x+y}^2=\sigma_{x-y}^2=\sigma_x^2+\sigma_y^2\).

4. From properties 2 and 3 we can deduce very useful property, which are very popular in statistics. Let\( X_1\),\(X_2\), …,\(X_N\) independent identically distributed random variables with same standard deviation σ, then standard deviation of the average value of \( X_1\),\(X_2\), …,\(X_N\)is next:

\(\sigma_{\overline{x}}=\frac{\sigma}{\sqrt{N}}\),

where \(\overline{X}=\frac{1}{N}\sum_{i=1}^NX_{i}\).

Consider one very popular applied aspect of using standard deviation for normal distribution. For this we consider normal distribution with parameters (μ, \(σ^2\)), where μ – mean of this distribution and \(σ^2\) – variance of this distribution. Density for this distribution have the next form:

\(f(x; μ, σ) = \frac{e\frac{(x-\mu)^2}{{2a}^2}}{\sqrt{2\pi\sigma}}\)

Then have the 68-95-99.7% rule for this distribution. This rule follows in mathematics notations: